Question: Factor the following expression: $2$ $x^2+$ $13$ $x+$ $21$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(2)}{(21)} &=& 42 \\ {a} + {b} &=& & & {13} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $42$ and add them together. The factors that add up to ${13}$ will be your ${a}$ and ${b}$ When ${a}$ is ${7}$ and ${b}$ is ${6}$ $ \begin{eqnarray} {ab} &=& ({7})({6}) &=& 42 \\ {a} + {b} &=& {7} + {6} &=& 13 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {2}x^2 +{7}x +{6}x +{21} $ Group the terms so that there is a common factor in each group: $ ({2}x^2 +{7}x) + ({6}x +{21}) $ Factor out the common factors: $ x(2x + 7) + 3(2x + 7) $ Notice how $(2x + 7)$ has become a common factor. Factor this out to find the answer. $(2x + 7)(x + 3)$